### A construction of the Gleason space

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We prove an analogue to Dordal’s result in P.L. Dordal, A model in which the base-matrix tree cannot have cofinal branches, J. Symbolic Logic 52 (1980), 651–664. He obtained a model of ZFC in which there is a tree $\pi $-base for ${\mathbb{N}}^{*}$ with no ${\omega}_{2}$ branches yet of height ${\omega}_{2}$. We establish that this is also possible for ${\mathbb{R}}^{*}$ using a natural modification of Mathias forcing.

Let G be a countably infinite group. We show that for every finite absolute coretract S, there is a regular left invariant topology on G whose ultrafilter semigroup is isomorphic to S. As consequences we prove that (1) there is a right maximal idempotent in βG∖G which is not strongly right maximal, and (2) for each combination of the properties of being extremally disconnected, irresolvable, and nodec, except for the combination (-,-,+), there is a corresponding regular almost maximal left invariant...

We introduce the notion of a coherent $P$-ultrafilter on a complete ccc Boolean algebra, strengthening the notion of a $P$-point on $\omega $, and show that these ultrafilters exist generically under $\U0001d520=\U0001d521$. This improves the known existence result of Ketonen [On the existence of $P$-points in the Stone-Čech compactification of integers, Fund. Math. 92 (1976), 91–94]. Similarly, the existence theorem of Canjar [On the generic existence of special ultrafilters, Proc. Amer. Math. Soc. 110 (1990), no. 1, 233–241] can...

This note establishes that the familiar internal characterizations of the Tychonoff spaces whose rings of continuous real-valued functions are complete, or $\sigma $-complete, as lattice ordered rings already hold in the larger setting of pointfree topology. In addition, we prove the corresponding results for rings of integer-valued functions.

We show that, under CH, the corona of a countable ultrametric space is homeomorphic to ${\omega}^{*}$. As a corollary, we get the same statements for the Higson’s corona of a proper ultrametric space and the space of ends of a countable locally finite group.